STFC STP Intro Summer School

1
Coronal seismology
Robert Erdélyi Robertus_at_sheffield.ac.uk SP2RC,
Department of Applied Mathematics, The
University of Sheffield (UK) http//robertus.sta
ff.shef.ac.uk
2
The Outline

• Introduction
• Observations of MHD waves
• Linear and (some) non-linear MHD waves
• Selected topics (stratification, thin flux
  tubes, autosolitons, applications)
• Conclusions

             
 
3
Why to bother Big questions

• What is the basis of stability and dynamics of
  solar atmospheric and ST structures?
• What mechanisms are responsible for heating in
  the solar atmosphere up to several million K?
• What accelerates the solar wind up to measured
  speeds exceeding 700 km/s?
• What are the physical processes behind the
  enormous energy releases (e.g. flares,
  substorms)?

4
Why to bother Big questions
Coronal heating Energy source for coronal
heating kinetic energy of convection zone
movie Goran Scharmer/SVST
Current sheets ?
Waves ?
Magnetic reconnection ?
image Neal Hurlburt/Karel Schrijver
image Marcus Aschwanden
image Dana Berry/NASA
5
What are MHD waves?

• How do we communicate in MHD? MHD is kind!

• MHD waves are propagating perturbations of
  magnetic field, plasma velocity and plasma mass
  density, described by the MHD (single fluid
  approximation) set of equations, which connects
  the magnetic field B, plasma velocity v ,
  kinetic pressure p and density ?.
• Non-relativistic approximation

             
 
6
Why to study MHD waves?

• MHD waves are believed to play a crucial role in
  the dynamics and structure of the solar interior,
  in the entire solar atmosphere (sunspots,
  chromosphere, TR, corona, solar wind) and in
  Earth magnetosphere. MHD waves are associated
  with
• the evolution and development of plasma
  perturbations,
• the transfer of plasma energy and momentum,
• plasma heating / acceleration,
• helioseismology, solar atmospheric (coronal)
  seismology, magnetosphere seismology.
• Also, we use it because simply they are there
  and affect us!

             
 
7
Magnetic coupling dynamic STS

• Photosphere chromosphere TR corona
  (inluding solar wind) magnetosphere Earths
  upper atmosphere are all magnetically coupled.
• Very highly structured and dynamic.
• MHD seismology is a perfect tool to study this
  coupled, dynamic an structured system.
• Two (biassed) particularly exciting aspects
• Influence of atmosphere on global oscillations.
• Role of p modes in the dynamics of the
  atmosphere! (Not yet explored.)

 
8
Structured solar atmosphere
The corona is highly structured in magnetic
field, in plasma density and in temperature.

• There are two main classes of coronal structures
  
• Closed structures loops (R100-200 Mm) which
  are hot ( 2-3x106 K) and dense (up to 7x1015
   m-3). Life time hours-days. However, loop
  ensembles called active regions (ARs) can live
  much longer.
• Open structures coronal holes, streamers,
  plumes inside the holes. Life time days-weeks.
• In addition, there are very dynamic plasma jets
  of various scales and speeds (erupting
  prominences, EEs, TRBs, etc.).

9
Do we expect solar MHD waves?

• Corona has frozen in field condition
• Magnetic field rooted into turbulent photosphere
• Generates waves that dump energy in corona
• Alfvén/(Slow) Magneto-acoustic waves

10
Do we expect solar MHD waves?
2D-simulation of a flux tube embedded in
photospheric granulation (radiation-MHD)
Steiner et al. (1997) ApJ 495, 468
Dutch Open Telescope, La Palma 12. Sept. 1999
Sütterlin Rutten
? 25 000 km x 38 000 km
observation in G-Band ? 430 nm granulation (Ø
?1000 km) G-band bright points small
magnetic flux tubes, which are brighter than
their surrounding
             
? 2400 km x 1400 km, ? 18 min
 
11
Do we expect solar MHD waves?
Rutten, R., ASP-CS, 184, 181, 1999
             
Low atmosphere role of underlying driver
 
12
Do we see MHD waves?
Before SOHO and TRACE MHD waves and
oscillations have been observed for a long time
in radio and optic bands Prominence
oscillations Periodic velocity and intensity
oscillations with various periods e.g. 1 hour,
3-5 min, 30 s. (They are seen from the Earth).
Radio pulsations Several periodicities were
detected in the MHD band by the analysis of the
coronal radio-emission. (See, e.g. Aschwanden
1987 for a review.) Roberts et al. (1983) Type
IV radio events have been observed and
interpreted as fast waves trapped in loops. The
idea of coronal seismology has been suggested for
the first time. EUV oscillations Probably, the
first observations of MHD waves in the corona
were reported by Chapman et al. (1972) with GSFC
extreme-ultraviolet spectroheliograph on OSO-7
(spatial resolution was few arcsec, cadence time
was 5.14 s). Mg VII, Mg IX and He II emission
intensity periodicities at about 262 s have been
detected.

13
Chromosphere filaments

• Hydrogen alpha filter image
• Thickness ? 2500 km

14
Do we see MHD waves?
Before SOHO and TRACE MHD waves and
oscillations have been observed for a long time
in radio and optic bands Prominence
oscillations Periodic velocity and intensity
oscillations with various periods e.g. 1 hour,
3-5 min, 30 s. (They are seen from the Earth).
Radio pulsations Several periodicities were
detected in the MHD band by the analysis of the
coronal radio-emission. (See, e.g. Aschwanden
1987 for a review.) Roberts et al. (1983) Type
IV radio events have been observed and
interpreted as fast waves trapped in loops. The
idea of coronal seismology has been suggested for
the first time. EUV oscillations Probably, the
first observations of MHD waves in the corona
were reported by Chapman et al. (1972) with GSFC
extreme-ultraviolet spectroheliograph on OSO-7
(spatial resolution was few arcsec, cadence time
was 5.14 s). Mg VII, Mg IX and He II emission
intensity periodicities at about 262 s have been
detected.
15
Do we see MHD waves?
Before SOHO and TRACE (ctd) Antonucci et al.
(1984) using Harvard College Observatory EUV
spectroheliometer on Skylab have detected
oscillations in the C II, O IV, and Mg X emission
intensity with periods of 117 s and 141 s. Soft
X-ray oscillations Harrison (1987) with Hard
X-ray Imaging Spectrometer on SMM have detected
soft X-ray (3.5-5.5 keV) pulsations of period 24
min (for six hours). Moreton-waves
Large scale wave motions have been discovered in
the corona in 1960!
16
Do we see MHD waves?
Moreton waves

• Seen in H? in the chromosphere at 10000 K
  (Moreton 60)
• Propagation speeds 450-2000 km/s, away from a
  flare site
• Propagate almost isotropically confined to an
  arc rarely exceeding 120º
• Have been identified as the intersection of
  coronal shock waves (due to a flare) with the
  chromosphere (Uchida 68 74)
• Are not seen to decelerate
• The generation mechanism has not been made clear
  yet

17
SOHO

• The Solar and Heliospheric Observatory
• Joint ESA and NASA project
• Suit of 12 instruments
• Launched in 1995
• 1.5 million km towards the Sun

18
Yohkoh TRACE

• Yohkoh (Sunbeam)
• Japan/UK/USA Mission
• Observed Sun in X-ray
• Launched in 1992

• Transition Region and Coronal Explorer
• NASA Small Explorer
• EUV Mission
• Incredible resolution

19
CLUSTER, RHESSI Hinode

• CLUSTER II
• Four satellite
• 3D magnetosphere
• July August 2000

• RHESSI
• Solar flare X-ray mission
• March 2001

• Hinode/Solar-B
• Japan/UK/USA Mission
• Successor of Yohkoh
• September 2006

20
Do we see MHD waves?
Surfing magnetic loops

• Example Rapid (every 15 s) TRACE 171 Angstrom
  image
• Track changes in brightness
• Wave travels outwards from B to T

21
Do we see MHD waves?
Surfing magnetic loops

• Difference image in brightness out from the loop
  base

22
Do we see MHD waves?
Compressive waves in solar plumes
             
 
23
Do we see MHD waves?
Compressive waves in solar plumes DeForest
Gurman (1998) and Ofman et al. (1999) with
SOHO/EIT and TRACE have detected and investigated
wave motions in polar plumes. Main properties of
these waves Outwardly propagating perturbations
of the intensity (plasma density) at 1.01-1.2 
R?, Quasiperiodic groups of 3-10 periods,
Periods about 10-15 min, The duty cycle is
roughly balanced, Speeds are about 75-150 km/s,
Amplitude (in density) is about 2-4  of the
background and grows with height. Ofman et al.
(1997) using white light channel (WLC) of the
SOHO/UVCS have detected density fluctuations in
coronal holes with periods ? 9 min at 1.9 R?.
             
 
24
Do we see MHD waves?
Compressive waves in long loops Berghmans
Clette (1999), with SOHO/EIT have observed
compressive propagating disturbances in coronal
loops (on the disk). Main findings Upwardly
propagating perturbations of the intensity
(plasma density) (very similar to the plume case,
but on the disc), With speed about 65-165 km/s,
Amplitude is ? 2 in intensity (? 1  in
density), The height growth of the amplitude has
not been found, No manifestation of downward
propagation. Travelling along almost all loops
analysed. Similar waves are observed with TRACE.
             
 
25
Do we see MHD waves?
26
Do we see MHD waves?
SOHO/TRACE examples (mainly TR and higher)
27
Do we see MHD waves?
Post-flare loop oscillations Aschwanden et al.
(1999) and Nakariakov et al (1999) with TRACE
have observed and investigated decaying kink-like
oscillations of coronal loops, excited by a
nearby flare (by a coronal Moreton wave?).
Main properties Oscillations are quasi-periodic
with periods of several minutes (256 s), Initial
displacement amplitudes are about several Mm for
loop radii about 100 Mm, Decay time about 14.5
min.
             
 
28
Do we see MHD waves? - Moreton waves
             
 
29
Do we see MHD waves? - Moreton waves
             
 
30
Do we see MHD waves?
New Coronal Moreton waves Thompson et al. (1999)
with SOHO/EIT have investigated a global coronal
wave generated by the coronal mass ejection or a
flare and occupying a significant part of the
solar disk. This wave has been called a coronal
Moreton wave. Properties accumulated from
observations of more than 50 events (see
http//umbra.nascom.nasa.gov/bjt/lscd/ or Ballai
et al. 2005 for details) The waves prefer to
propagate radially from the epicentre, stopping
at neutral lines and coronal hole boundaries, and
distorted by active regions. Speeds range is
from 200-600 km/sec. Active regions distort the
waves locally, bending them possibly toward the
lower Alfvén speed regions. The waves can cause
"visible deflection" of coronal magnetic field
lines and probably are associated with filament
oscillations.
31
Do we see MHD waves? Coronal Moreton waves
32
Do we see MHD waves? Coronal Moreton waves

• Moreton waves on difference images after solar
  eruption

33
Do we see MHD waves? X-ray waves

• Seen by Yohkoh/SXT propagating in the corona
• Interpreted as coronal MHD fast-mode weak shock
  (Narukage et al. 02)
• Propagation speed of 630100 km/s
• Believed to be the coronal counterpart of
  chromospheric Moreton waves (?)

             
 
34
Do we see MHD waves?
Moreton waves ? X-ray waves

• 1997.11.03 NOAA AR 8100
• Both propagate in the same direction and agree
  in location
• X-ray waves are well correlated to Moreton waves
• X-ray waves are the coronal counterpart of the
  Moreton waves (Narukage et al 02)

             
 
35
Do we see MHD waves? Sun quakes

• Solar quakes high energy electrons slam the
  solar surface

36
Do we see MHD waves? Sun quakes

• Solar quakes high energy electrons slam the
  solar surface

37
Do we see MHD waves? Sun quakes

• Solar quakes high energy electrons slam the
  solar surface

38
Do we see MHD waves?
Non-thermal broadening of coronal emission
lines Most probably associated with MHD waves).
Measured broadening of minor ion spectral lines
is formed by two effects, thermal broadening and
non-thermal broadening associated with the
Doppler shift due to unresolved line-of-sight
motions
where Ti is the temperature of the line forming
ion, k is the Boltzmann constant , vLOS is the
line-of-sight (LOS) velocity, 2/3ltalt1
Non-thermal broadening of the UV and EUV coronal
lines has been known for 25 years from Skylab.
             
 
39
Do we see MHD waves?
Non-thermal broadening of coronal emission lines
(ctd) Recent findings Ofman Davila (1997)
using SOHO/UVCS measured unresolved motions with
speeds up to  300 km/s at about 1.7  R?. Erdélyi
et al. (1998) using SOHO/SUMER found that the
non-thermal center-to-limb LOS velocity increases
from few km/s to almost 100 km/s in coronal
loops. Banerjee et al. (1998) using SOHO/SUMER
found that the non-thermal LOS velocity increases
from 27 km/s at 20 Mm above the limb to 46 km/s
at 62 Mm. Chae et al. (1998), SOHO/SUMER, LOS
velocities of 20-30 km/s on the disc Esser et al.
(1999), SOHO/UVCS, LOS velocities of 20-23 km/s
at 1.35-2.1  R? There is some discrepancy in
results found using different instruments.
However, the results clearly show the presence of
the unresolved line-of-sight plasma motions
caused by waves and/or turbulence in the corona.
40
Do we see MHD waves?- Solar spicules

• Solar spicules are thin, hair-like jets of gas
  seen on the solar limb in chromospheric emission
  lines
• They occur predominantly at supergranule
  boundaries and appear to be guided along the
  intense magnetic flux tubes gathered there
• Typical properties are
• Some spicules display rapid rotation about their
  axis, typically of the order of 25km s-1
• The spicule rise is probably not ballistic,
  although the evidence for this is not conclusive

SOHO Image of the Solar limb taken March 96
             
Ha Image from the Big Bear Solar Observatory,
California
 
41
Do we see MHD waves?- Solar spicules

- 1.0 Å

- 0.8 Å

- 0.6 Å

- 0.4 Å

- 0.2 Å

Ha centre
(BBSO)

• reaching heights of up to 10 000 km
• almost vertical (but not quite)
• apparent vertical motions of 20 km/s
• forming bushes at network
• what drives spicules?
• mass supply to corona and solar wind?
• spicules - mottles - moss ?
• relation to UV-Spicules?

42
Do we see MHD waves?
Solar tornadoes (May be connected with MHD
waves). Pike Mason (1998) with SOHO/CDS
Macrospicule-like (a jet) features have been
identified in the polar regions both on the limb
and disk. Blue- and red-Doppler-shifted emission
occur on either side of the feature axis,
indicating the presence of rotation (called solar
tornado). The rotation velocities increase with
height.
             
 
43
Atmospheric seismology
Oscillations ubiquitous in Sun

• Solar atmosphere
• More local oscillations
• Sunspot oscillations, prominence oscillations,
  coronal loop oscillations, plume oscillations
• EIT waves?

• Solar interior
• Global oscillations
• p/f/g-modes

• Unifying feature of variety of solar atmospheric
  oscillations
• Waveguide concept
• MHD description

44
Atmospheric seismology
Oscillations ubiquitous in solar atmosphere

• Higher atmosphere
• TR, corona
• Magnetic environment

• Lower atmosphere
• Ph, Ch, possibly TR
• Isolated flux tubes
• Effect of stratification

vA
vA
Stratification leads to the Klein-Gordon effect
Roberts (1981), Rae Roberts (1982)
45
Atmospheric seismology

• What is the motivation?
• Source of atmospheric heating solar
  wind/particle acceleration
• Understand atmospheric structures (spicules,
  prominences, loops, plumes, etc.)

Wave properties (speed, amplitude, spectrum)
spectroscopic
Atmospheric diagnostic parameters (temperature,
density)
Observations
Geometric properties of waveguides (structuring,
shape, curvature)
imaging
Atmospheric physical parameters (B, fine
structure, transport coefficients)
46
Linear theory of MHD waves

• Static/steady stationary background
• Superimpose linear motions on this background
• Write physical quantities as
• f(r,t)f0(r)f1(r,t) f1/ f0 ltlt1
• Reduce full set of nonlin PDEs of MHD to a set
  of ODEs
• Choice initial value problem, boundary value
  problem, eigenvalue problem
• Eigenvalue problem of linear waves/oscillations
  exp(i?t)

             
 
47
Linear theory of MHD waves
Linearised ideal MHD equations
48
Linear MHD waves in uniform plasma

• No characteristic length scale defined by the
  equilibrium
• Constant equilibrium magnetic field , e.g.

• Superposition of linear waves
• exp(ikxx ikyy ikzz), k(kx, ky, kz)wave
  vector

49
Linear MHD waves in uniform plasma
Alfvén speed
Characteristic speeds
sound speed
50
Linear MHD waves in uniform plasma
Consider dynamics of perturbations of this
stationary state. In the linear limit, the set of
MHD equation splits into two uncoupled
subsets (i) for the variables Vy and By (Alfvén
wave) (ii) and for ?, p, Vx, Vz and Bx
(magnetoacoustic waves)

51
Linear MHD waves in uniform plasma
Consider dynamics of perturbations of this
stationary state. In the linear limit, the set of
MHD equation splits into two uncoupled
subsets (i) for the variables Vy and By (Alfvén
wave) (ii) and for ?, p, Vx, Vz and Bx
(magnetoacoustic waves)
52
Linear MHD waves in uniform plasma
Alfvén waves

• Properties
• Transverse oscillation driven by magnetic
  tension forces
• Does not perturb density ? incompressible (in
  linear limit)
• Cant propagate across field lines
• Group velocity (d?/dk) is along B0

53
Linear MHD waves in uniform plasma
Alfvén waves

• Properties
• Transverse oscillation driven by magnetic
  tension forces
• Does not perturb density ? incompressible (in
  linear limit)
• Cant propagate across field lines
• Group velocity (d?/dk) is along B0

54
Linear MHD waves in uniform plasma
Alfvén waves

• Properties
• Transverse oscillation driven by magnetic
  tension forces
• Does not perturb density ? incompressible (in
  linear limit)
• Cant propagate across field lines
• Group velocity (d?/dk) is along B0

55
Linear MHD waves in uniform plasma
Alfvén waves

• Properties
• Transverse oscillation driven by magnetic
  tension forces
• Does not perturb density ? incompressible (in
  linear limit)
• Cant propagate across field lines
• Group velocity (d?/dk) is along B0

56
Linear MHD waves in uniform plasma
Alfvén waves
When B0z0 there can be two linearly polarized
plane Alfvén waves, one perturbing Vy, By and the
other Vx, Bx.
For harmonic perturbations exp(i?t -kz)
combination of two linearly polarized waves gives
us elliptically polarized Alfvén waves
57
Linear MHD waves in uniform plasma
Alfvén waves
The vector of the magnetic field perturbation
rotates along an ellipse at the x,y-plane. When A
B, the wave is circularly polarized, with
Bconst. Circularly polarized Alfvén waves
(even of finite amplitude) are an exact solution
of the ideal MHD equations for a uniform plasma
58
Linear MHD waves in uniform plasma
Magnetoacoustic waves
Harmonic perturbations
Dispersion relation for MAW
DR bi-quadratic? slow and fast magnetoacoustic
waves
59
Linear MHD waves in uniform plasma
Magnetoacoustic waves
The polar plot for phase speeds (?/k) for case
ßlt1
60
Linear MHD waves in uniform plasma
Magnetoacoustic waves
The polar plot for group speeds (d?/dk)
61
Linear MHD waves in uniform plasma
Slow waves

• Properties
• Anisotropic wave propagation largely confined to
  magnetic field
• Driven by magnetic pressure and tension forces
• Does perturb density/pressure
• Cant propagate across field lines

62
Linear MHD waves in uniform plasma
Slow waves

• Properties
• Anisotropic wave propagation largely confined to
  magnetic field
• Driven by magnetic pressure and tension forces
• Does perturb density/pressure
• Cant propagate across field lines

63
Linear MHD waves in uniform plasma
Slow waves

• Properties
• Anisotropic wave propagation largely confined to
  magnetic field
• Driven by magnetic pressure and tension forces
• Does perturb density/pressure
• Cant propagate across field lines

64
Linear MHD waves in uniform plasma
Fast waves

• Properties
• Roughly isotropic wave propagation
• Driven by magnetic pressure and tension forces
• Does perturb density/pressure
• Propagates fastest perpendicular to B

65
Linear MHD waves in non-uniform plasma

• Characteristic length scale defined by the
  inhomogeneity
• Equilibrium quantities are functions of position
• Continuum of resonant Alfvén and slow waves
• Discrete slow and fast modes discrete Alfvén
  modes
• Efficient damping in non-ideal MHD
• MHD waves with mixed character and wave
  transformation

66
Linear MHD waves in non-uniform plasma

• Properties of MHD waves depend upon the angle
  between the wave vector and the magnetic field ?
  waves "feel" the direction of the field.
• When the magnetic field is not straight, Alfvén
  and slow waves should follow the field, because
  they are confined to the field.
• Even when the field is straight, inhomogeneities
  in the field absolute value, density and pressure
  affect the characteristic speeds of the waves
  (the Alfvén and the sound speeds) and,
  consequently, affect the waves.
• ? Guided propagation of MHD waves, linear
  coupling of different MHD modes, phase mixing of
  Alfvén waves, resonant absorption, appearance of
  wave dispersion, etc.

67
Linear MHD waves in non-uniform plasma
68
Linear MHD waves in non-uniform plasma
Total pressure balance
Characteristic speeds

• Alfvén speed
• Sound speed
• Tube (cusp) speed

69
Linear MHD waves in non-uniform plasma
Fourier transform in homogeneous directions (y, z)
Boundary conditions at fixed x ? Dispersion
Relation
70
Linear MHD waves in non-uniform plasma
Consider d/dy0, though Vy, By ?0 (i.e. 2.5D)
Alfvén modes perturbing Vy, By
Magnetoacoustic modes perturbing Vx, Vz, Bx, Bz,
?
71
Linear MHD waves in non-uniform plasma
Magnetoacoustic modes are governed by
B.C.seigenvalue problem. Eigenfunctions define
transversal (x) structure of waves eigenvalues
define dispersion for waves.
Alfvén

Singularities
resonances!
Cusp
72
Linear MHD waves in non-uniform plasma
Magnetoacoustic modes
Evanescent solutions modes or trapped or guided
(or ducted) waves Dispersion is determined by
the ratio of the longitudinal wavelength to the
characteristic spatial scale of inhomogeneity.

• The modes can have different structures in x
  direction (inhomogeneity), which allows us to
  classify them
• kink and sausage modes (perturbing or not
  perturbing the structure axis, respectively)
• body and surface modes (oscillating or
  evanescent inside the structure, respectively,
  and both evanescent outside the structure)

73
Linear MHD waves in non-uniform plasma
Magnetoacoustic modes
             
 
74
Linear MHD waves in non-uniform plasma
Magnetoacoustic modes
In addition, different modes of the same parity
and transversal structure can be distinguished as
slow and fast modes. Different modes have
different properties dispersion relations,
characteristic speeds, excitation conditions and
observational manifestation.
             
 
75
MHD waves in magnetic tubes and slabs (structured
plasma)
76
MHD waves in magnetic tubes and slabs (structured
plasma)

• Linear motions of a compressible
  cylindrical/slab plasma

77
MHD waves in magnetic tubes and slabs (structured
plasma)

• Flux tube/slab in static equilibrium
• Cylindrical coordinates r,f, z x?r, y ? f, z
  ?z.
• Magnetic surfaces in tube cylinders rconst.
• Perturbations proportional to exp(im fikzz) m,
  kz azimuthal and axial wave numbers in tube
• Magnetic field B0B0ez, BeBeez

78
MHD waves in magnetic tubes and slabs (structured
plasma)

• Pressure balance at rx (x?a)

• External and internal solutions of the MHD
  equations have to be matched by the boundary
  conditions the total pressure continuity,

• the normal velocity continuity,

• and the condition of the mode localization,

• Similar boundary conditions are in the slab
  case.

79
MHD waves in magnetic tubes and slabs

• Dispersion relation for slabs

• where a is the slab semi-width and the tanh/coth
  terms correspond to the sausage/kink modes,
  respectively.

• DRs describe both surface (m02gt0) and body
  (m02lt0) waves.
• In ALL cases me2lt0 (non-leaky waves)

80
MHD waves in magnetic tubes and slabs

• Dispersion relation for tubes

m02gt0 ? surface waves
Roberts (1981), Edwin Roberts (1983)
m02 -n02 lt0 ? body waves. Note n0
refers to sausage, n1 to kink modes, etc.
81
MHD waves in magnetic tubes and slabs

• The number n determines the mode structure

• Therefore, the modes with n0 are sausage modes
  and with n1 are kink, ngt1 flute modes.

             
 
82
Theory of tube oscillations
Solutions to DR for lower atmosphere magnetic
tubes
Torsional waves
n0

• Main modes
• (Alfvenic) torsional (incompressible)
• fast sausage (B, ?)
• fast kink (almost incompressible)
• slow (acoustic) type (?, v)

n1
Edwin Roberts (1983)
83
Theory of tube oscillations
Solutions to DR for coronal loops
n0

• Main modes
• fast sausage (B, ?)
• fast kink (almost incompressible)
• (Alfvenic) torsional (incompressible)
• slow (acoustic) type (?, v)

n1
84
Theory of tube oscillations
Solutions to DR for closed tubes
Expected oscillation periods in coronal loops

• Main modes
• sausage
• kink
• (Alfvenic) torsional
• slow (acoustic) type

sausage modes P 0.1 ? 5 s
B? gt 0, ?? gt 0
kink modes P 1.4 ? 14 min
?? ? 0
V??
Alfvenic modes P 1 s
?? 0
Vz?
slow modes P 7 ?70 min
B? gt 0, ?? lt 0
KG
Aschwanden (2003), Wang (2004)
85
The Klein-Gordon effect
Stratified atmosphere (gconst)

• Equilibrium
• Scale height
• 1D, sound waves
• Introduce

Webb Roberts, Sol. Phys, 56, 5
(1978) Ulmschneider and cos, series of papers in
AA
86
The Klein-Gordon effect
Isothermal atmosphere
(acoustic cut-off frequency)
Photosphere ?ac 4.8 mHz ? P 210 s
Corona ?ac 0.18 mHz ? P 91.7 min

• Leakage of photospheric motion into LA
• Sound, slow, Alfvén waves
• ?5/3 ? 1
• Non-adiabatic plasma
• Inclination of magnetic wave guides

Review by Roberts (2003)
87
The Klein-Gordon effect
Stratified atmosphere (VALIIIC, gconst)
 
88
Atmospheric seismology
Moss oscillations
89
Atmospheric oscillations

• Early (i.e. pre-SOHO/TRACE) non-imaging
  observations (mainly TR and higher)
• Visible (Koutchmy et al. 1983)
• Radio (Aschwanden, 1987)
• EUV (Chapman et al. 1972)
• Soft X-rays (Jakimiec Jakimiec, 1974)

Culgoora radio spectograph

• Radio observations
• Periods P0.01- 1000 s

0307 UT on 1972 May 16
McLean Sheridan (1973) Radio pulsations at 230
MHz, period of P 4.28 0.01 s
90
Atmospheric oscillations
SOHO/TRACE examples (mainly TR and higher)
91
Atmospheric oscillations
SOHO/TRACE examples (mainly TR and higher)

• Two types of observed oscillations can be
  distinguished
• Propagating waves (Ofman et al. 1997, DeForest
  Gurman 1998, Berghmans Clette 1999, De Moortel
  et al. 2000, 2002a,b,c, Robbrecht et al. 2001,
  King et al. 2003, Marsh et al. 2003)
• Standing waves

i) Standing kink-mode oscillations by TRACE
(Aschwanden et al. 1999, 2002, Nakariakov et al.
1999, Schrijver Brown 2000, Schrijver et al.
2002, )
ii) Standing slow-mode oscillations by
SOHO/SUMER (Kliem et al. 2002 Wang et al. 2002,
2003a,b)
Reviews by Aschwanden (2003), Wang (2004)
92
Resonant absorption

• Inhomogeneous plasmas natural behaviour
• Easy wave energy transfer resulting in heating
• Condition to occur ?driver ?local
• Versatile as could/may/viable to explain
• - local/atmospheric heating
• - power loss of acoustic waves in sunspots
• - damping of helioseismic (p/f/g) eigenmodes
• - energisation of MHD waves in
  magneto/heliosphere

             
 
93
The Concept of Resonant Absorption

• Global modes resonantly interact with local MHD
  modes

• Dissipation

Steady state

• Damping of global oscillations

             
 
94
The Concept of RA
?driver ?local
Ideal MHD equations singular ? dissipation ?
heating Connection Formulae
             
 
95
Remember Observational facts
             

• Example of inhomogeneity coronal loops

 
96
Still remember Observations
             
 
97
Observationals
             

• Background flows

 
98
Space Atmosphere Research Center
Nonlinear resonant MHD waves in stratified
plasmas

• Importance of nonlinearity ? Ruderman et al.
  1997ab, Ballai et al. 1998ab, Erdélyi et al. 2001
• Nonlinearity slightly reduces the efficiency of
  coupling
• Nonlinearity generates mean flows
• Nonlinear correction p2/8

Result nonlinearity does not seem to be important
             
 
99
What we really need?

• Observations, observations, observation
• Indirect observations (e.g. Erdélyi et al. 98)
• Evidence for resonant waves (e.g. mean flow)
• Observe reconnection driven (resonant) MHD waves
  
• SOHO JOP 122 probably some other JOPs

             
 
100
Model improvement
2D static magnetic model of the transition
region ? pressure, Lorentz, gravity, heat flux,
rad.loss, heating Gabriel (1976), Phil. Trans.
A281, 339
101
Model improvement
(De Pontieu, Tarbell, Erdélyi, ApJ 590, 502, 2003)
102
Conclusions

• There are MHD waves in the Sun, in STP
• MHD waves are natural for plasma
  heating/acceleration
• MHD waves are sensitive to flows
• Must take into account effects of
  inhomogeneity/structuring. The structuring plays
  a crucial role in the wave dissipation and
  transformation.
• The waves are an efficient tool for MHD
  seismology of the solar atmosphere, which allows
  us to determine values of the mean parameters of
  the corona, such as the magnetic field strength,
  density, pressure, and transport coefficients.
  Some of these values the magnetic field
  strength, viscosity, resistivity and thermal
  conductivity, are not open to measurement by any
  other means. We can do this by measuring the
  properties of MHD waves and oscillations
  (periods, wavelengths, amplitudes, temporal and
  spatial signatures), combined with theoretical
  modelling of the wave phenomena.

             
 
103
The end